Unfortunately, although this feature is frequently requested, this is not really possible with the transit evaluator and Add GTFS to a Network Dataset toolbox, as mentioned in the "Limitations and weaknesses" section of the User's Guide. This problem is because of the way the Network Analyst solvers work under the hood. They do not have the ability to separately track and limit different states (walking vs. transit).
For the Service Area Lines output only, you can kind of hack it to sort of do what you're asking. You should continue to use your transit travel time cost attribute as the impedance attribute. But, additionally, you should set the accumulate attributes to accumulate a walk time attribute. For streets, the walk time attribute should use the same evaluator calculation as the transit travel time attribute. However, for transit lines and the stop-street connectors, you should set the evaluator to a constant of 0. You could also choose to make this attribute calculate walking distance instead of time. When you accumulate this attribute, it will add time or distance whenever the streets are traversed (where the traveler is walking), but it will not add any impedance for travel along the transit lines and connectors. Then, you can eliminate all output Lines where the accumulated walk time or distance exceeds your threshold.
